Chapter 7 Sets & Probability Section 7.6 Bayes’ Theorem

Tree Diagram for Two Events, E and F

Bayes’ Theorem (Special Case – Two Events)

Bayes’ Theorem Three or more events)

A personal computer manufacturer buys 38% of its chips from Japan and the rest from America. 1.7% of the Japanese chips are defective, and 1.1% of the American chips are defective. a.) Find the probability that a chip is defective and made in Japan. b.) Find the probability that a chip is defective and made in America. c.) Find the probability that a chip is defective. d.) Find the probability that a chip is defect-free.

The University of Metropolis requires its students to pass an examination in college-level mathematics before they can graduate. The students are given three chances to pass the exam; 61% pass it on their first attempt. 63% of those that take it a second time pass it then; and 42% of those that take it a third time pass it then. a.) What percent of the students pass the exam? b.) What percent of the students are not allowed to graduate because their performance on the exam? c.) What percent of the students take the exam at least twice? d.) What percent of the students take the test three times?

Suppose the space shuttle has three separate computer control systems – the main system and two backup duplicates of it. The first backup would monitor the main system and kick in if the main system failed. Similarly, the second backup would monitor the first. We can assume that a failure of one system is independent of a failure or another system, since the systems are separate. The probability of failure for any one system on any one mission is known to be 0.01. a.) Find the probability that the shuttle is left with no computer control system on a mission. b.) How many backup systems does the space shuttle need if the probability that the shuttle is left with no computer control system on a mission must be 1 in 1 billion?

Medical researchers have recently devised a diagnostic test for “white lung” (an imaginary disease caused by the inhalation of chalk dust). Teachers are particularly susceptible to this disease; studies have shown that half of all teachers are afflicted with it. The test correctly diagnoses the presence of white lung in 99% of the persons who have it and correctly diagnoses its absences in 98% of the persons who do not have it. a.) Find the probability that a teacher whose test results are positive actually has white lung. b.) Find the probability that a teacher whose test results are negative does not have white lung. c.) Find the probability that a teacher whose test results are positive does not have white lung. (False positive) d.) Find the probability that a teacher whose test results are negative actually has white lung. (False negative)

A cable television firm buys television remote control devices from three different manufacturers: one in Korea, one in Singapore, and one in California. Forty-two percent of its remote controls are purchased from the Korean manufacturer, 23% are purchased from the Singapore manufacturer, and the rest are purchased from the California manufacturer. Three percent of the Korean remote controls are defective, 7.1% of the Singapore remote controls are defective, and 1.9% of the California remote controls are defective. a.) What percent of the defective remote controls are made in Korea? b.) What percent of the defective remote controls are made in Singapore? c.) What percent of the defective remote controls are made in California?

Three cards are dealt from a deck of 52 Three cards are dealt from a deck of 52. Find the probability of each of the following. a.) All three are hearts. b.) Exactly two are hearts. c.) At least two are hearts. d.) The first is an ace of hearts, the second a two of hearts, and the third a three of hearts.